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The wave equation

$$u_{xx}(x,t)=\frac {1}{c^2}u_{tt}(x,t) $$

requires two initial conditions because the equation is second order:

IC1: $$u(x,0)= f(x)$$

IC2: $$u_{t}(x,0)= g(x)$$

But when it is factored:

$$u_{tt} - c^2 u_{xx} = \bigg( \frac{\partial }{\partial t} - c \frac{\partial }{\partial x} \bigg) \bigg( \frac{\partial }{\partial t} + c \frac{\partial }{\partial x} \bigg) u(x,t) = 0$$

it only requires one initial condition when each of the factors is set equal to zero:

$$\bigg( \frac{\partial }{\partial t} - c \frac{\partial }{\partial x} \bigg) u(x,t)= 0 $$ $$\bigg( \frac{\partial }{\partial t} + c \frac{\partial }{\partial x} \bigg)u(x,t)=0 $$

IC1: $$u(x,0)= f(x)$$

because the 'factors' are first order.

So I understand mathematically why in the first case two ICs are needed whereas in the second case only one IC is needed. But if the factored and unfactored wave equations are equivalent--containing all of the same information-- , I don't understand intuitively or physically the difference in the number of required ICs.

My question is: Intuitively and physically, why does the wave equation need two ICs when the 'factored' wave equation needs only one?

See also:

Intuition into why the wave equation needs the second initial condition (e.g. velocity)

Intuitively, why are only two initial conditions needed for the wave equation? Why not 3 or 4?

https://math.stackexchange.com/q/2706776/147776

https://physics.stackexchange.com/a/403761/45664

EDIT 6/2/18 SEE @jcandy ANSWER BELOW FOR CORRECTIONS TO THIS QUESTION AND FOR THE ANSWER

The equations with the factors should have been written

$$\bigg( \frac{\partial }{\partial t} - c \frac{\partial }{\partial x} \bigg) u(x,t)= v $$ $$\bigg( \frac{\partial }{\partial t} + c \frac{\partial }{\partial x} \bigg)v(x,t)=0 $$

with an IC given for each equation.

user45664
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1 Answers1

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Your factorization is not correct. Think about the simpler equation $\partial_{xx} u = 0$. This can be "factored" according to $\partial_x \left( \partial_x u\right) = 0$. One would then say that the factored system can be written as \begin{align} \partial_x v &~= 0 \\ \partial_x u &~= v \end{align} where $v$ is an auxiliary function. In each line above, the derivative accounts for one undetermined coefficient. Thus, there are a total of two initial conditions required. A more detailed explanation is here: https://math.stackexchange.com/a/84268/275678

jcandy
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